2. INTRODUCTORY LECTURE 3 Lecture 3: Analysis of Lab Work Electricity and Measurement (E&M)BPM – 15PHF110

3. In This Lecture… 1. Objectives 2. Multiple Measurement Uncertainty 3. Standard Deviation 4. Random and Systematic Uncertainty 5. Combining Uncertainties in Calculations

4. Analysing Data - Objectives  Explain how to produce the uncertainty in a set of multiple readings using a Normal (Gaussian) Distribution and the Standard Deviation,  calculate the mean, standard deviation and standard error in the mean,  Understand the difference between,  the standard deviation of a sample from a population and  the standard deviation of the entire population.  Be able to contrast how uncertainty effects the accuracy and precision of the data collected,  linking accuracy to systematic uncertainty,  linking precision to random uncertainty.  Combine uncertainties during calculations to provide the overall uncertainty in the calculated value,  when adding or subtracting data,  when multiplying and dividing data,  when power terms, etc. are involved.

5. More simply… How do we calculate the mean How do we calculate the Standard Deviation How do we calculate the Standard Error How do we combine errors where we have separate errors in different readings We will do the basics Practice on this Using your calculator

6. Readingd /cm 13.4 23.9 33.2 42.7 53.5 6 7 83.9 93.0 103.2 113.7 123.4 133.9 143.4 153.5 Reducing Uncertainty With Multiple Measurements Usually written as Readingd /cm 13.4 23.9 33.2 42.7 43.5 5 6 73.9 83.0 93.2 103.7 113.4 123.9 133.4 143.5

7. Reducing Uncertainty With Multiple Measurements Usually written as Is the symbol for “sum the following”, i.e. add them all together

8. 3.03.33.63.9 Spread of Data Readingd /cm 13.4 23.9 33.2 43.5 5 6 73.9 83.0 93.2 103.7 113.4 123.9 133.4 143.5    * * * * * * * # # # # Gaussian or Normal Distribution

9. Standard Deviation and Normal Distributions Standard Deviation shown on the normal distribution. More Precise Less Precise 68% of measured values (green dots) are within the blue zone

10. Standard Deviation and Uncertainty

11. Two Types of Standard Deviation If n is the total population under test we use this version. Excel formula: =STDEV.P(specify range here) (Population standard deviation) If n is a sample from a much larger population we use this version – for physics experiments this one is usually best. Excel formula: =STDEV.S(specify range here) (Sample standard deviation) Use this one

12. Uncertainty and Normal Distributions Readingd /cm 13.4 23.9 33.2 43.5 5 6 73.9 83.0 93.2 103.7 113.4 123.9 133.4 143.5 0.3 fractional percentage

13. Example, Find the standard error in the mean for the following measurements: 12345 5.04.94.75.35.2 Standard deviation, Standard error in the mean, Mean, ? ??

14. Recording Data Where possible, make multiple measurements to determine the random error properly and pick up any mistakes. Where this is not possible, regard any single measurement as one drawn from a Gaussian (normal) population.  Use common sense and experience to estimate what the uncertainty would be – you are looking to accommodate 2/3 of all possible measurements within the ± spread you quote. Usually this means ± half the… last digit on a display, division marking on a scale, the range of a fluctuating needle 2.31 Reading = 2.310 ±0.005 s Reading = 39.0 ±0.5mm Reading = 6.5 ±0.5A

15. Summary of Random and Systematic Random Equally likely to give results that are above and below the true value, cancelling each other out to give a zero expected value. Random errors are present in all experiments and therefore the researcher should be prepared for them. Measured values Systematic Biases in measurement resulting in the mean of the separate measurements differing significantly from the true value, e.g. calibration or zero error. They can be either constant, or related (e.g. proportional or a percentage) of the true value. True Value

16. Summary of Random and Systematic Not accurate Not precise - large systematic error - large random error Not accurate Precise - large systematic error - small random error Accurate Not precise - small systematic error - large random error Accurate Precise - small systematic error - small random error True Value Measured Value True Value Measured Value True Value Measured Value True Value Measured Value

17. Combining Uncertainties When combining measurements in an equation these two ways of quoting uncertainties become very important… Adding or Subtracting the uncertainty in the result is found from the, standard uncertainties of the initial values. standard uncertainties Multiplying or Dividing the uncertainty in the result is found from the, relative uncertainties of the initial values. relative uncertainties

18. Combining Uncertainties: Addition and Subtraction

19. Combining Uncertainties: Multiplication and Division From

20. Combining Uncertainties: Multiplication and Division cont.… Same relative uncertainty Different standard uncertainty Multiplication Division

21. Combining Uncertainties: Power Terms And The Rest… (Used rarely so no need to learn these, look them up when required)

22. Combining Uncertainties: Insignificant Uncertainty

23. Practice Makes Perfect Using a calculator: =AVERAGE(B2:B6)0.561 7 =STDEV.S(B2:B6)0.011 8 =B8/SQRT(A6)0.005 9 AB 1 10.560 2 20.575 3 30.565 4 40.560 5 50.545 6

24. Practice Makes Perfect 10.560 20.575 30.565 40.560 50.545 0.561 0.011 0.005

25. Practice Makes Perfect 10.560 20.575 30.565 40.560 50.545 0.561 0.011 0.005

26. Processing uncertainties - summary  Explain how to produce the uncertainty in a set of multiple readings using a Normal (Gaussian) Distribution and the Standard Deviation,  calculate the mean, standard deviation and standard error in the mean,  Understand the difference between,  the standard deviation of a sample from a population and  the standard deviation of the entire population.  Be able to contrast how uncertainty effects the accuracy and precision of the data collected,  linking accuracy to systematic uncertainty,  linking precision to random uncertainty.  Combine uncertainties during calculations to provide the overall uncertainty in the calculated value,  when adding or subtracting data,  when multiplying and dividing data,  when power terms, etc. are involved.